Real numbers can be thought of as points on an infinitely long
number line

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th-century mathematics. The current standard axiomatic definition is that real numbers form the unique Dedekind-completeordered field(R ; + ; · ; up to an isomorphism,[a] whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent.

The reals are uncountable; that is: while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers: the cardinality of the set of all real numbers (denoted
c
{\displaystyle {\mathfrak {c}}}
and called cardinality of the continuum) is strictly greater than the cardinality of the set of all natural numbers (denoted
ℵ
0
{\displaystyle \aleph _{0}}
'aleph-naught'). The statement that there is no subset of the reals with cardinality strictly greater than
ℵ
0
{\displaystyle \aleph _{0}}
and strictly smaller than
c
{\displaystyle {\mathfrak {c}}}
is known as the continuum hypothesis (CH). It is known to be neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC), the standard foundation of modern mathematics, in the sense that some models of ZFC satisfy CH, while others violate it.

The real number system
(
R
;
+
;
⋅
;
<
)
{\displaystyle (\mathbf {R} ;{}+{};{}\cdot {};{}
can be defined axiomatically up to an isomorphism, which is described hereafter. There are also many ways to construct "the" real number system, for example, starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another possibility is to start from some rigorous axiomatization of Euclidean geometry (Hilbert, Tarski, etc.) and then define the real number system geometrically. From the structuralist point of view all these constructions are on equal footing.

The last property is what differentiates the reals from the rationals (and from other, more exotic ordered fields). For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational.

These properties imply Archimedean property (which is not implied by other definitions of completeness). That is, the set of integers is not upper-bounded in the reals. In fact, if this were false, then the integers would have a least upper bound Ngreen Chicca Shoulder Women Borse 8880 Bag Green OwRYOqT; then, N – 1 would not be an upper bound, and there would be an integer n such that n > N – 1, and thus n + 1 > N, which is a contradiction with the upper-bound property of N.

The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields R1 and R2, there exists a unique field isomorphism from R1 to R2, allowing us to think of them as essentially the same mathematical object.

The sum and the product of two non-negative real numbers is again a non-negative real number, i.e., they are closed under these operations, and form a positive cone, thereby giving rise to a linear order of the real numbers along a number line.

The real numbers make up an infinite set of numbers that cannot be injectively mapped to the infinite set of natural numbers, i.e., there are uncountably infinitely many real numbers, whereas the natural numbers are called countably infinite. This establishes that in some sense, there are more real numbers than there are elements in any countable set.

Real numbers can be used to express measurements of continuous quantities. They may be expressed by decimal representations, most of them having an infinite sequence of digits to the right of the decimal point; these are often represented like 324.823122147..., where the ellipsis (three dots) indicates that there would still be more digits to come. This hints to the fact that we can precisely denote only a few, selected real numbers with finitely many symbols.

More formally, the real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by non-zero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that, if a non-empty set of real numbers has an upper bound, then it has a real least upper bound. The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound (e.g. 1.5) but no (rational) least upper bound: hence the rational numbers do not satisfy the least upper bound property.

A main reason for using real numbers is that the reals contain all limits. More precisely, a sequence of real numbers has a limit, which is a real number, if (and only if) its elements eventually come and remain arbitrarily close to each other. This is formally defined in the following, and means that the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). :

A sequence (xn) converges to the limitx if its elements eventually come and remain arbitrarily close to x, that is, if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − x| is less than ε for n greater than N.

Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.

The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2).

The completeness property of the reals is the basis on which calculus, and, more generally mathematical analysis are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.

can be made arbitrarily small (independently of M) by choosing N sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing that
e
x
{\displaystyle e^{x}}
is well defined for every x.

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.

First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant.

Additionally, an order can be Dedekind-complete, as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.

These two notions of completeness ignore the field structure. However, an Multicolor3 Dunland Boy Backpack Pupil Diaper Shield Bag Multicolor3 Cartoon wrxwF0g (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the previous section Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterization of the real numbers.) It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.

But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.

As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.

Every nonnegative real number has a square root in R, although no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make R the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra.

The fieldR of real numbers is an extension field of the field Q of rational numbers, and R can therefore be seen as a vector space over Q. Zermelo–Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: there exists a set B of real numbers such that every real number can be written uniquely as a finite linear combination of elements of this set, using rational coefficients only, and such that no element of B is a rational linear combination of the others. However, this existence theorem is purely theoretical, as such a base has never been explicitly described.

Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are (non-"standard") elements of the set of the real numbers (rather than being elements of an extension thereof, as in Robinson's theory).

The continuum hypothesis posits that the cardinality of the set of the real numbers is
ℵ
1
{\displaystyle \aleph _{1}}
; i.e. the smallest infinite cardinal number after
ℵ
0
{\displaystyle \aleph _{0}}
, the cardinality of the integers. Paul Cohen proved in 1963 that it is an axiom independent of the other axioms of set theory; that is: one may choose either the continuum hypothesis or its negation as an axiom of set theory, without contradiction.

In set theory, specifically descriptive set theory, the Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals".

Mathematicians use the symbol R, or, alternatively, ℝ, the letter "R" in blackboard bold (encoded in Unicode as U+211DℝDOUBLE-STRUCK CAPITAL R (HTML &#8477;)), to represent the set of all real numbers. As this set is naturally endowed with the structure of a field, the expression field of real numbers is frequently used when its algebraic properties are under consideration.

In mathematics, Bag Laptop Casual Bag For Messenger Zonlin Ladies Burgundy Bags Shoulder Women real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number (as in "the set of all reals").

The long real line pastes together ℵ1* + ℵ1 copies of the real line plus a single point (here ℵ1* denotes the reversed ordering of ℵ1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ1 in the long real line but not in the real numbers. The long real line is the largest ordered set that is complete and locally Archimedean. As with the previous two examples, this set is no longer a field or additive group.

^More precisely, given two complete totally ordered fields, there is a unique isomorphism between them. This implies that the identity is the unique field automorphism of the reals that is compatible with the ordering.